Cell bulging and extrusion in a three-dimensional bubbly vertex model for curved epithelial sheets

Abstract

Cell extrusion is an essential mechanism for controlling cell density in epithelial tissues. Another essential element of epithelia is curvature, which is required to achieve complex shapes, like in the lung or intestine. Here we introduce a three-dimensional bubbly vertex model to study the interplay between extrusion and curvature. We find a generic cellular bulging instability at topological defects which is much stronger than for standard vertex models. Analyzing cell shapes in three-dimensional imaging data of spherical mouse colon organoids, we infer that pentagonal cells have an increased basal interfacial tension, suggesting that cells at topological defects react to the different force conditions. Using the bubbly vertex model, we show that such basal tensions stabilize against the predicted instability and result in better cell shape control than tissue-scale mechanisms such as lumen pressure and spontaneous curvature. Our theory suggests that epithelial curvature naturally leads to bulged and extrusion-like cell shapes because the interfacial curvature of individual cells at the defects strongly amplifies buckling effected by tissue-scale topological defects in elastic sheets. Our results highlight the complex interplay of forces across scales in three-dimensional tissue organization.

Figures (12)

• Figure 1: Cell bulging at topological defects in the bubbly vertex model. (a) Schematic depiction of the vertex model (VM) representation of a cell with apical, basal and lateral faces via an interpolation point of the polygonal faces, and of the bubbly vertex model (BVM) representation with curved interfaces. (b) Spherical VM and BVM shell after energy minimization with random initialization of cell centers. BVM shell is after minimization in Surface Evolver. Number of cell neighbors is color-coded. (c) Opening angles of the cells by neighbor number for the VM and BVM, where the half angle is determined for each lateral interface (inset) and then averaged over lateral interfaces. Violin plots show distribution with the horizontal line marking the mean value. Tensions $\Gamma_\mathrm{a}=\Gamma_\mathrm{b}=0.9$, $\Gamma_\mathrm{l}=1$. (d) Mean opening angles of cells with different neighbor numbers (color-coded; red pentagons, black hexagons, blue heptagons) in different BVM-shells with random initial cell configurations ($N=6$) and varying apical and basal tensions $\Gamma_\mathrm{a}=\Gamma_\mathrm{b}$ and $\Gamma_\mathrm{l}=1$. All shells with $400$ cells.
• Figure 2: Experimentally observed cell shapes at topological defects in mouse colon (mCol) organoids. (a) Microscopy image of spherical mouse colon organoid. Green is membrane ($\beta$-catenin) and magenta is F-actin. The luminal side is apical (high actin) and the outer side basal (facing the matrix). (b) The organoid is segmented in 3D and the number of neighbors (color-coded) is determined from the basal polygonal edges. (c) For this organoid the opening angles $\delta/2$, averaged over all lateral interfaces, vary for different neighbor numbers. Each dot represents one cell, the bars depict the means. (d) The relative angle increase at pentagonal and heptagonal cells, compared to hexagonal cells. Every data point is one organoid ($N=22$). (e) Microscopy image of organoid treated with Latrunculin A (LatA). (f) Examples of pentagonal and hexagonal cells as segmented in 3D. (g) Opening angles for organoid treated with LatA. (h) Comparing the relative angle increase $\rho$ from wildtype ($N=13$) and LatA-treated samples ($N=17$), we find larger effects at topological defects with LatA. Scale bars in (a) and (e) are $20\,\mu\mathrm{m}$. Stars indicate: $*\ p<0.05$, $**\ p<0.01$, $***\ p<0.001$. Tests against $0$ in (d) with Wilcoxon and against each other in (h) with Mann-Whitney-U tests.
• Figure 3: Surface tension inference in mouse colon (mCol) organoids. (a) Assuming force balance, the relative tensions can be inferred from the dihedral angles of the interfaces. For the reconstructed organoid the outer (basal) tensions are shown. (b,c) For each wildtype (WT) (b) and Latrunculin A treated (LatA) (c) organoid we determine the difference of the averaged inferred tensions $\hat{\Gamma}$ for $n$-gons from hexagons for apical (a) and basal (b) interfaces. The averaged cell-specific apico-basal tension difference is also shown (a-b). (d) Pentagonal basal tension deviations versus relative angle increase of pentagons compared to hexagons $\rho(5)$ for WT and LatA samples. Solid line is linear regression, which intersects a tension deviation of $0.0$ at $\rho(5)\approx0.17$. (e) Relative mid-plane area deviations of pentagons (red) and heptagons (blue) from hexagons for WT and LatA organoids. (f) Pentagonal basal tension deviations versus relative area deviations of pentagons for WT and LatA samples. Solid line is linear regression, which intersects a tension deviation of 0.0 at $\left(\langle A\rangle_5-\langle A \rangle_6\right)/\langle{A}\rangle_6=-0.015$. Pearson correlation coefficient in (d,f) is $\rho_\mathrm{c}$. Stars indicate: $*\ p<0.05$, $**\ p<0.01$, $***\ p<0.001$. Tests against $0$ in (b,c,e) with Wilcoxon test and against each other in (e) with Mann-Whitney-U-test.
• Figure 4: Modulation of instability via different mechanisms explored through the BVM. (a) The shell from Fig. \ref{['fig:fig_shells_pentagonal_collapse']} simulated with additional effects: pressurized with $V/V_\mathrm{VM}=2.0$, rigidified with $\kappa_\mathrm{if}=0.7$, and pentagonal basal (outer) defect tension of $\Gamma_\mathrm{b}^{(5)}=0.957$. (b) Opening angles of pentagonal (red), hexagonal (black) and heptagonal cells (blue) as a function of the number of simulated cells. (c) Opening angles for different apico-basal tensions differences $\Delta\Gamma=\Gamma_\mathrm{a}-\Gamma_\mathrm{b}$, corresponding to different spontaneous curvatures. (d) Opening angles for different luminal volume constraints, where $V/V_\mathrm{VM}\approx1$ corresponds to the case without a luminal volume constraint. (e) Opening angles for rigidified shells different interfacial bending rigidities $\kappa_\mathrm{if}$, where $\kappa_\mathrm{if}=0$ is the regular BVM. (f) Dependence on the basal apical tensions $\Gamma^{(5)}=\Gamma_\mathrm{a}^{(5)}=\Gamma_\mathrm{b}^{(5)}$ of pentagonal cells, with unchanged tensions in other cells. (g) Dependence on basal (outer) tension $\Gamma_\mathrm{b}^{(5)}$ of pentagonal cells which is changed compared to the outer tension of the other cells. All other tensions are unchanged compared to the reference. Markers are average angles and shaded regions display $5\%$ to $95\%$ quantiles. Single red lines in (f) and (g) are individual pentagonal cells and dotted vertical lines mark the reference case.
• Figure 5: Lowered buckling threshold for icosahedral instability in the bubbly vertex model (BVM) compared to the vertex model (VM). (a,b) Icosahedral shells with Caspar Klug indices $(8,0)$ with tension $\Gamma=0.6$ in the VM (a) and the BVM (b). (c) Asphericity $\alpha$ in the BVM-shell (symbols) and of the continuum curve describing the VM (taken from Refs. Lidmar_PRE_2003_icosahedral_instability_shellsDrozdowski_PRR24_Topological_defects_VM_shells). Inset shows shell with size $(2,0)$, $\Gamma=0.6$. Radii are rescaled by the Föppl-von Kármán number and with a correction $k_\mathrm{ico}$ from non-linear elasticity. (d) For a small shell (Caspar Klug indices $(2,0)$) the average ratio of luminal and outer interfacial area in the BVM (solid) and VM (dashed). Inset shows pentagonal cell with collapsed luminal interface ($\Gamma=1.0$). Pentagonal cells are shown in red.